Størmer regions for axisymmetric magnetic multipole fields

Abstract

The motion of a single charged particle in the space outside of a compact region of steady currents is investigated. The charged particle is assumed to produce negligible electromagnetic radiation, so that its energy is conserved. The source of the magnetic field is represented as a point multipole. After a general description, attention is focused on magnetic fields with axial symmetry. Lagrangian dynamical theory is utilized to identify constants of the motion as well as the equations of motion themselves. In particular, the energy and the canonical angular momentum conjugate to the azimuthal angle are conserved for axial symmetry. Following the precedent of some authors, and for brevity, the constant canonical angular momentum conjugate to the azimuthal angle will be called Størmer’s integral. In the present work, the magnetic vector potential equivalent to a given spherical harmonic scalar potential is also found. The qualitative method of Størmer is then used to examine charged particle motion in axisymmetric multipole fields of all orders, and while the treatment is fully relativistic, it also encompasses nonrelativistic motion. This qualitative method divides the configuration space of the charged particle into sets in which motion is either forbidden or allowed, i.e., it permits a topological examination of dynamical motion. The results obtained here allow for the determination of trapped-particle regions around those astrophysical objects and spacecraft plasma propulsion systems that have an associated axisymmetric magnetic field described by a multipole of given order.